Chapter 1.1, Problem 11E

(a)

To determine

The most general conditions on the subsets $A$ and $B$ of $U$ under which the equality $A\cap B=A$ holds.

(b)

To determine

The most general conditions on the subsets $A$ and $B$ of $U$ under which the equality $A\cup B\text{'}=A$ holds.

(c)

To determine

The most general conditions on the subsets $A$ and $B$ of $U$ under which the equality $A\cup B=A$ holds.

(d)

To determine

The most general conditions on the subsets $A$ and $B$ of $U$ under which the equality $A\cap B\text{'}=A$ holds.

(e)

To determine

The most general conditions on the subsets $A$ and $B$ of $U$ under which the equality $A\cap B=U$ holds.

(f)

To determine

The most general conditions on the subsets $A$ and $B$ of $U$ under which the equality $A\text{'}\cap B\text{'}=\varphi$ holds.

(g)

To determine

The most general condition on the subset $A$ of $U$ under which the equality

$A\cup \varphi =U$ holds.

(h)

To determine

The most general condition on the subset $A$ of $U$ under which the equality

$A\text{'}\cap U=\varphi$ holds.